Problem: All of the 4th grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$6.00$ each for teachers and $$4.50$ each for students, and the group paid $$63.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$24.00$ each for teachers and $$12.50$ each for students, and the group paid $$197.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6x+4.5y = 63}$ ${24x+12.5y = 197}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-24x-18y = -252}$ ${24x+12.5y = 197}$ Add the top and bottom equations together. $ -5.5y = -55 $ $ y = \dfrac{-55}{-5.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {6x+4.5y = 63}$ to find $x$ ${6x + 4.5}{(10)}{= 63}$ $6x+45 = 63$ $6x = 18$ $x = \dfrac{18}{6}$ ${x = 3}$ You can also plug ${y = 10}$ into $ {24x+12.5y = 197}$ and get the same answer for $x$ ${24x + 12.5}{(10)}{= 197}$ ${x = 3}$ There were $3$ teachers and $10$ students on the field trips.